^{1}

^{2}

This paper proposes the combined Laplace-Adomian decomposition method (LADM) for solution two dimensional linear mixed integral equations of type Volterra-Fredholm with Hilbert kernel. Comparison of the obtained results with those obtained by the Toeplitz matrix method (TMM) demonstrates that the proposed technique is powerful and simple.

The Volterra-Fredholm integral equation (V-FIE) arises from parabolic boundary value problems. The integral equations appear in many problems of physics and engineering. The Adomian decomposition method (ADM) was proposed by George Adomian in [

We consider the linear (V-FIE) with singular kernel given by

There are several techniques that have been utilized to handle the integral Equation (1) in [

In this work, we display numerical techniques to obtain numerical solution for linear mixed integral equation with Hilbert kernel. In Section 2, we talk about the existence and uniqueness of the solution. In Section 3, we discuss the Adomian decomposition method, as one of the well known technique and we note that the Adomian polynomials do not appear in this work because we handle linear problems. In Section 4, we present the Laplace Adomian decomposition method and apply this method to linear mixed integral equation with Hilbert kernel. In Sections 5 and 6, we display the Toeplitz matrix method.

Consider the integral Equation (1), the functions

and the time t,

In order to guarantee the existence of a unique solution of Equation (1) we assume through this work the following conditions:

(i) The kernel of position

Satisfies the discontinuity condition

(ii) The kernel of time

(iii) The given function

nuous in the space

(iv) The unknown function

Adomian decomposition method [

where the components

The polynomials

Substituting Equation (2) into Equation (1) to get

The components

We consider the kernel

Laplace transform to both sides of Equation (1) gives:

The linear term

The Adomian decomposition method introduces the recursive relation

Applying the inverse Laplace transform to the first part of Equation (9) gives

In this part, a numerical technique is used, in the integral Equation (1) to obtain a system of linear integral equa-

tions with singular kernel, so we divide the interval

where

where

In this section, we apply (TMM) to obtain the numerical solution of the SFIEs (10) with singular kernel, each equation in this system can be written in a simplify form

The integral term in Equation (12) can be written as

We approximate the integral in the right hand side of Equation (13) by

where

We can, clearly solve the result set of two equations for

where

Hence, Equation (13) takes the form

where

The integral Equation (12), after putting

The formula (18) represents a linear system of algebraic equation , where u is a vector of

The matrix

The solution of the system (18) can be obtained in the form

The error term

Example 1: Consider the linear mixed integral equation with Hilbert kernel

we obtain

In this paper, we applied (LADM) for solution two dimensional linear mixed integral equations of type Volterra- Fredholm with Hilbert kernel. Additionally, comparison was made with Toeplitz matrix method (TMM). It could be concluded that (LADM) was an effective technique and simple in finding very good solutions for these sorts of equations.

Using Maple 18, we obtain

2.340000000E−11 | −5.877852312E−04 | 1.620000000E−11 | −5.877852676E−04 | −5.877852514E−04 | −2.5132E+00 | |

1.110000000E−11 | −9.510565048E−04 | 6.200000000E−12 | −9.510565109E−04 | −9.510565171E−04 | −1.2566E+00 | |

5.307017821E−12 | 5.307017821E−12 | 1.760000000E−11 | 1.760000000E−11 | 0.000000000E+00 | 0.0000E+00 | |

1.110000000E−11 | 9.510565270E−04 | 6.200000000E−12 | 9.510565217E−04 | 9.510565155E−04 | 1.2566E+00 | |

1.004000000E−10 | 5.877851542E−04 | 1.620000000E−11 | 5.877852392E−04 | 5.877852554E−04 | 2.5132E+00 | |

6.262300000E−07 | −1.763293141E−02 | 4.368700000E−07 | −1.763399441E−02 | −1.763355754E−02 | −2.5132E+00 | |

2.985300000E−07 | −2.853139695E−02 | 1.668800000E−07 | −2.853152893E−02 | −2.853169551E−02 | −1.2566E+00 | |

1.433100000E−07 | 1.433104100E−07 | 5.399280000E−07 | 5.399279998E−07 | 0.00000000E+00 | 0.0000E+00 | |

2.994600000E−07 | 2.853199494E−02 | 1.668700000E−07 | 2.853186233E−02 | 2.853169546E−02 | 1.2566E+00 | |

2.710200000E−06 | 1.763084744E−02 | 4.368700000E−07 | 1.763312079E−02 | 1.763355766E−02 | 2.5132E+00 | |

8.208891800E−03 | −4.032407864E−01 | 5.509811800E−03 | −4.169594878E−01 | −4.114496760E−01 | −2.5132E+00 | |

3.788791000E−03 | −6.619507701E−01 | 2.183584500E−03 | −6.63559775E−01 | −6.657395620E−01 | −1.2566E+00 | |

1.906364353E−03 | 1.906364353E−03 | 6.859341229E−03 | 6.859339549E−03 | 0.000000000E+00 | 0.0000E+00 | |

4.001864900E−03 | 6.697414260E−01 | 2.055721600E−03 | 6.677952824E−01 | 6.657395608E−01 | 1.2566E+00 | |

3.479475530E−02 | 3.766549229E−01 | 5.588835400E−03 | 4.058608434E−01 | 4.114496788E−01 | 2.5132E+00 |

The authors would like to thank the King Abdulaziz city for science and technology.

M. Salgueiro,P. González,T. F. Pena,J. C. Cabaleiro,Fatheah Ahmed Hendi,Manal Mohamed Al-Qarni, (2016) Comparison between Adomian’s Decomposition Method and Toeplitz Matrix Method for Solving Linear Mixed Integral Equation with Hilbert Kernel. American Journal of Computational Mathematics,06,177-183. doi: 10.4236/ajcm.2016.62019